Assume that \(G\)  is either isomorphic to \(PSL_{3}(3)\)  or to \(PSL_{3}(3):\left\langle \sigma \right\rangle \) , where \(\sigma \)  is of order 2, and that \(k=12\) . If \((x,B)\)  is any flag of \(\mathcal {D}\) , then either \(G_{x} \cong F_{39}\) , or \(G_{x} \cong F_{78}\) , respectively, by [1]}.
Moreover \(b=12\cdot 13\cdot \lambda \)  and the cyclic subgroups of order 3
lying in \(G_{x}\)  belong to the conjugacy class \(\mathcal {K}_{2}\)  defined
above. Hence \(\left|G_{B}\right|=\frac{36c}{\lambda }\) , where \(c=\left[ G:PSL_{3}(3)\right] \) . Then \(\lambda \mid 3c\) , as \(12\mid \left|G_{B}\right|\) , with \(\lambda \geqslant 2\) . Thus, \(\left( \lambda ,c\right) =(2,2),(3,1)\) , \((3,2)\)  or \((6,2)\) .

\(\left( \lambda ,c\right) \ne (2,2)\) .

Assume that \(\left( \lambda ,c\right) =(2,2)\) . Then \(G\cong PSL_{3}(3):\left\langle \sigma \right\rangle \) , \(\left|G_{B}\right|=36\) , and \(G_{x,B}\cong Z_{3}\) , with \( G_{x,B} \in \mathcal {K}_{2}\) . Then the point-\(G_{x}\) -orbits on \(\mathcal {D}\)  are \(\left\lbrace  x\right\rbrace  \) , one of length 13, two ones
of length 26 and two ones of length 39 by [2]}. Then \(B\)  intersects
each of these orbits in \(1,1,2,2,3,3\)  points respectively by Lemma REF (3). Hence, \(G_{x,B}\)  fixes at least six points on \(B\) . On the other hand,
Lemma REF (2) implies \(\left[ N_{G}(G_{x,B}):N_{G_{x}}(G_{x,B})\right]=6 \) . Therefore, \(\left|\mathrm {Fix(}G_{x,B}\mathrm {)}\right|=6\) .
Thus, \(\mathrm {Fix(}G_{x,B}\mathrm {)}\subset B\)  and hence the number of
blocks of \(\mathcal {D}\)  containing \(\mathrm {Fix(}G_{x,B}\mathrm {)}\)  is at most 2, as \(\lambda =2\) . Thus either \(r=\left|\mathcal {K}_{2}\right|\)  or \(r=2\left|\mathcal {K}_{2}\right|\)  and we reach a contradiction, as \(r=24\)  but \(\left|\mathcal {K}_{2}\right|=312\) .

If \(\left( \lambda ,c\right) =(3,1)\) , then \(\mathcal {D}\)  is
isomorphic to the 2-design constructed in Example REF (1).

Assume that \(\left( \lambda ,c\right) =(3,1)\) . Then \(G\cong PSL_{3}(3)\)  and \(G_{B}\)  acts regularly on \(B\) . Let \(H\)  be a Sylow 3-subgroup of \(G_{B}\) . By
Lemma REF (1)–(2), either \(H\in \mathcal {K}_{1}\)  and \(G_{B}\cong D_{12}\) , or \(H\in \mathcal {K}_{2}\)  and \(G_{B}\cong A_{4}\) , according to whether \(H\) 
is or is not normal in \(G_{B}\)  respectively. With the aid of [3]} we
see that, no flag-transitive 2-designs occur with \(G_{B}\cong D_{12}\) , whereas, up to
isomorphism, only one flag-transitive 2-\((12^{2},12,3)\)  design occurs with \(G_{B}\cong A_{4}\) . Thus, such a 2-design is
necessarily isomorphic to that constructed in Example REF (1).

\(\left( \lambda ,c\right) \ne (3,2)\) .

Assume that \(\left( \lambda ,c\right) =(3,2)\) . Then \(G\cong PSL_{3}(3):\left\langle \sigma \right\rangle \) , \(G_{x,B}\cong Z_{2}\)  and \(r=39\) . Since \((G^{\prime })_{x}\cong F_{39}\)  by [1]}, it follows that \((G^{\prime })_{x,B}=1\) . Therefore, \((G^{\prime })_{x}\)  acts transitively on
the set of blocks of \(\mathcal {D}\)  incident with \(x\) . Also, \(G^{\prime }\) 
acts point-transitively on \(\mathcal {D}\) , as \(G\)  acts point-primitively on \(\mathcal {D}\) . Thus, \(G^{\prime }\)  acts flag-transitively on \(\mathcal {D}\) .
Therefore, \(\mathcal {D}\)  is isomorphic to the 2-design constructed in Example REF (1) by (ii), and hence \(G^{\prime }\) 
is its full flag-transitive automorphism group.
However, this is impossible, since \(G\cong PSL_{3}(3):\left\langle \sigma \right\rangle \)  acts flag-transitively on \(\mathcal {D}\)  by our assumption.

If \(\left( \lambda ,c\right) =(6,2)\) , then \(\mathcal {D}\)  is
isomorphic to the 2-design constructed in Example REF (2).

If \(\left( \lambda ,c\right) =(6,2)\) , then \(G\cong PSL_{3}(3):\left\langle \sigma \right\rangle \) , \(r=78\) , \(G_{x,B}=1\)  and \(\left|G_{B}\right|=12\) . By using Lemma REF (1)–(2), it is easy to see that, either \(G_{B}\) 
is isomorphic to \(Z_{12}\) , \(D_{12}\)  (three conjugacy classes) or to \(A_{4}\) .
With the aid of [3]} we see that, no flag-transitive 2-designs occur with \(G_{B}\cong Z_{12}\)  or \(D_{12}\) , whereas, up to isomorphism, only one flag-transitive 2-\((12^{2},12,6)\)  design occurs with \(G_{B} \cong A_{4}\) . Thus, such 2-design is
necessarily isomorphic to that constructed in Example REF (2).
